All killers are not necessarily mentally ill.
Many are evil...pure and simple.
For example, many attribute other motives to terrorists besides evil. Ex: poverty, disenfranchisement, or in many people's mind like yours mental illness.
But much murder simply comes from evil desires.
Mass shooter Luke Woodham explained...
“I am not insane, I am angry. I killed because people like me are mistreated every day. I did this to show society, push us and we will push back. ... All throughout my life, I was ridiculed, always beaten, always hated. Can you, society, truly blame me for what I do? Yes, you will. ... It was not a cry for attention, it was not a cry for help. It was a scream in sheer agony saying that if you can’t pry your eyes open, if I can’t do it through pacifism, if I can’t show you through the displaying of intelligence, then I will do it with a bullet. “